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Abstract and Applied Analysis
Volume 2013, Article ID 321930, 15 pages
http://dx.doi.org/10.1155/2013/321930
Research Article

Stability and Global Hopf Bifurcation Analysis on a Ratio-Dependent Predator-Prey Model with Two Time Delays

1Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China
2Department of Mathematics and Information Science, Zhoukou Normal University, Zhoukou, Henan 466001, China

Received 4 September 2013; Revised 3 November 2013; Accepted 5 November 2013

Academic Editor: István Györi

Copyright © 2013 Huitao Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A ratio-dependent predator-prey model with two time delays is studied. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semitrivial equilibrium is addressed. By using the theory of functional equation and Hopf bifurcation, the conditions on which positive equilibrium exists and the quality of Hopf bifurcation are given. Using a global Hopf bifurcation result of Wu (1998) for functional differential equations, the global existence of the periodic solutions is obtained. Finally, an example for numerical simulations is also included.